On the almost everywhere convergence of multiple Fourier series of square summable functions

We prove that if the lacunary partial sums of the Fourier series of every square summable function concerning each one-dimensional orthonormal system $\Phi_{1},\dots,\Phi_{d}$ converge almost everywhere, then the product system $\Phi_{1}\times\cdots\times\Phi_{d}$ also has a similar property for a quite general type of partial sums.